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"... We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wa ..."

We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multi-wavelet expansions of a given order L.

"... In this paper, we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discrete-time by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of ..."

In this paper, we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discrete-time by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of orthogonal or biorthogonal warped wavelets by means of rational transfer functions. We show that the discrete-time warped wavelets lead to well-defined continuous-time wavelet bases, satisfying a warped form of the two-scale equation. The shape of the wavelets is not invariant by translation. Rather, the &quot;wavelet translates&quot; are obtained from one another by allpass filtering. We show that the phase of the delay element is asymptotically a fractal. A feature of the warped wavelet transform is that the cut-off frequencies of the wavelets may be arbitrarily assigned while preserving a dyadic structure. The new transform provides an arbitrary tiling of the time--frequency plane, which can be designed by selecting as little as a single parameter. This feature is particularly desirable in cochlear and perceptual models of speech and music, where accurate bandwidth selection is an issue. As our examples show, by defining pitch-synchronous wavelets based on warped wavelets, the analysis of transients and denoising of inharmonic pseudo-periodic signals is greatly enhanced.

"... We present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis. Using Green's Theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficient ..."

We present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis. Using Green&apos;s Theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product. The multidimensional filter coefficients are precomputed exactly as the solution of a two-scale relation. To demonstrate the performance improvement of the new method, we compare it with existing methods such as pixel-based approaches and approximation of the region by a polygon. We also propose an alternate scheme when the scaling function is sinc(x).

"... In this paper, we present a new algorithm for the design of orthonormal two-band rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedur ..."

In this paper, we present a new algorithm for the design of orthonormal two-band rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedure, which explains its exponential convergence and adaptability under various linear constraints (e.g., regularity). Although the filters obtained from this algorithm are suboptimally designed, they show excellent frequency selectivity.

"... The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smoot ..."

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”.

The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Q-factor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Q-factors (desirable for processing oscillatory signals) or the same low Q-factor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible ‘constant-Q’ discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). The wavelet can be made to resemble a Gabor function and can hence have good concentration in the timefrequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform’s redundancy and the flexibility allowed by frequency-domain filter design. I.

"... This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the t ..."

This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the time-frequency plane. A straightforward algorithm is described for the construction of minimal-length perfect reconstruction filters with a specified number of vanishing moments; whereas, in the non-redundant rational case, no such algorithm is available. The algorithm is based on matrix spectral factorization. The analysis/synthesis functions (discrete-time wavelets) can be very smooth and can be designed to closely approximate the derivatives of the Gaussian function.

"... Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for time-scale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter ba ..."

Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for time-scale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechies-type filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Gröbner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.

"... In this paper, we describe a new set of dyadic wavelet frames with three generators, ψi(t), i = 1, 2, 3. The construction is simple, yet the wavelets cover the time-frequency plane in an effective way: one of the three wavelets is exactly a half-integer shift of another [ψ3(t) = ψ2(t−0.5)], and the ..."

In this paper, we describe a new set of dyadic wavelet frames with three generators, ψi(t), i = 1, 2, 3. The construction is simple, yet the wavelets cover the time-frequency plane in an effective way: one of the three wavelets is exactly a half-integer shift of another [ψ3(t) = ψ2(t−0.5)], and the spectrum of the third wavelet is concentrated halfway between the spectrums of the first wavelet and its dilated version [Ψ1(ω) is concentrated between Ψ2(ω) and Ψ2(2 ω)]. This arrangement provides a higher sampling in both time and frequency, which leads to expansive wavelet transforms that are approximately shift-invariant and have intermediate scales. The wavelet frames presented in this paper are compactly supported and have vanishing moments.

"... In this paper we consider an extension of the wavelet transform leading to the construction of wavelets with arbitrary band-width. The new wavelets are complete, orthonormal and dyadic; nevertheless their bandwidth is not constrained to be one oc-tave, rather it may be designed by selecting a set of ..."

In this paper we consider an extension of the wavelet transform leading to the construction of wavelets with arbitrary band-width. The new wavelets are complete, orthonormal and dyadic; nevertheless their bandwidth is not constrained to be one oc-tave, rather it may be designed by selecting a set of parameters. The construction of the new bases starts in the discrete-time domain, exploiting properties of the Laguerre transform. Fur-thermore, we provide a procedure to define continuous-time warped wavelets. Flexibility of the bandwidth allocation allows for more and improved applications of the wavelet transform, such as signal coding, the design of auditory model based filter-banks and transient detection in pseudoperiodic signals, pointed out in the paper. 1.